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The Wikipedia page Extreme weather events of 535–536 lists under "Documentary evidence" that:

• Low temperatures, even snow during the summer (snow reportedly fell in August in China during the Northern and Southern dynasties, which
  caused the harvest there to be delayed


• "A dense, dry fog" in the Middle East, China and Europe

The page does provide a reference (Ochoa, George; Jennifer Hoffman; Tina Tin (2005). Climate: The Force That Shapes Our World and the Future of Life on Earth. Emmaus, PA: Rodale. p. 71.), but upon reading the referenced book, I noticed that the original quote was itself unsubstantiated, as shown in the picture I took below (at the end of the book there was only a Further Reading list):

I then proceeded to find other academic sources online. One article titled Spatial Response to Major Volcanic Events in or about AD 536, 934 and 1258 includes a passage quite similar to that found on the Wikipedia page, except slight variation in time:

The above results for Mongolia are consistent with the observations of
summer frosts and snow, crop failure, decreased star visibility and
famine documented for northern China in July and August of AD 536–7.

it listed the following citation for that passage:

Pang, K. and Chou, H.-H.: 1985, Unpublished research, in S. Weisbard,
‘Excavating Words: A Geological Tool’, Science News 127, 91–94.

So I went to this cited article, which contains the following:

The effect of this volcano was also felt in China. Pang reports that,
during the spring and fall equinoxes each year, the ancient Chinese
looked for Canopus -the brightest star in the constellation of Alpha
Carina-to assure themselves of good times ahead and to demark the
seasons. In A.D. 536, however, Canopus was not seen. Pang found
records from the state of Ching in southern China reporting frost and
snow in July and August that killed the seedling crop, causing a major
famine the following autumn. The accounts also show that the effects
of the eruption were noted until 538. Other kingdoms reported similar
disasters; one record indicated that the weather was so severe that 70
to 80 percent of the people starved to death, says Pang.

Unfortuantely, the article does not provide any specific references to this quote, and it seems that Pang's research was indeed unpublished.

I have just begun to look into the massive amount of Chinese chronicles from this period with the hope of finding the exact entries quoted by Pang, in particular the highlighted quote concerning the severity of the disaster, which seems like to be highly exaggerated. In the meantime, I would really appreciate if someone could point me to a specific passage from an ancient Chinese source that confirms aforementioned descriptions.

Thank you.

The chronicles Beishi "History of the North" and Nanshi "History of the South" have at least some of what you are looking for.

The Nanshi chronicle mentions frost and snow. This is cited in Catastrophe: An Investigation into the Origins of the Modern World by David Keys:

In July 537 China was hit by frost, while in August it snowed. The
History of the Southern Dynasties recorded that “in July in Qingzhou
and [another province] there was a fall of frost” and that “in August
in Qingzhou there was snow,” which “ruined the crops.”

The Nanshi chronicle is also cited in the The Palgrave Handbook of Climate History:

the early seventh-century Nanshi chronicle refers to “yellow dust”
that “fell like snow” in 536 and 537. In the latter year, it “filled
scoops when picked up.” The dust was almost certainly Gobi sand (not
volcanic ash), but this signals that 536 and 537 were unusually dry.

Keys also has the following paragraph which cites the Beishi:

“Because of drought, there was an imperial edict which ordered that in
the capital [Ch’ang-an], in all provinces, commanderies and districts,
one should bury the corpses,” says the Bei shi for late April to early
May 535.1 “[There was] great drought. [The government] had to provide
water at the city gates [of Ch’ang-an] and the hall gates [of the
palace] as well as the gates of the government offices,” says the
entry for late June and early July of the same year. Then in September
536, in the north Chinese “provinces of Bian, Si, Zhuo and Jian, hail
fell” and there was “a great famine.” By December the situation had
deteriorated to such an extent that the government had to send special
inspectors “to investigate [the conditions of] the famished refugees
who were roaming around north of the Yellow River.”

The next paragraph then adds:

And in Shaanxi province, “the land within the Passes,” The Annals of
the Western Wei in the Bei shi state that there was “a great famine,”
and that “the people practiced cannibalism and 70 to 80 percent of the
population died.”

This is the supplementary answer to that of Lars Bosteen.

As he said, according to this site( Japanese ), Nanshi records multiple quite unusual events occurred in Lyan Dynasty ( English ).

Quote ( from the above Japanese source )

534年閏12月丙午、西南方で雷鳴が二度、響いた。

>In the intercalary December, 534AD, we heard the lightening sound twice at South West. ( Nanshi 1.7.4.1 )

これは先にも引用した火山噴火への言及だが、恐らく梁の人々はこのようなことはすぐ忘れてしまっただろう。不可思議なことと言ってもやはり、庶民には何の関係もない･･･筈だった。しかしそれから異常なことが次々と起こった。翌535年の冬十月、雪のような黄塵が降った1。これが火山灰だったか気候の大異変に起因する季節はずれの砂嵐だったかは今となっては確かめる術もない。この黄塵は翌536年11月2、537年2月1日3、550年1月丁巳の日4にも観測された。

This would be the reference to the volcano explosion of the Krakatoa, in Indonesia, that occurred in 535 AD, though I assume people in Lyan Dynasty would've forgotten such lightenings soon from their memories. So those, the unusual lightening sounds, seemed.., should have been nothing to do with the living of people. But parallels of extraordinary climate events occurred multiple time since the sound. In the next year, in October, yellow dusts amounted like snow ( Nanshi 1.7.4.1 ). We have no clue now to confirm that this was the dust of the result of the volcano explosion that happened in the previous year. But these dusts are recorded in Nanshi, observed in the next year, in November 536 ( Nanshi 1.7.4.2 ), on the first of February 537, ( Nanshi 1.7.4.3 ), on 4th of January in 550 ( Nanshi, 1.7.6.4 ).

何かがおかしい。梁の民もそう思い始めたことだろう。しかし時既に遅し、大異変の歯車は回り始めていた。537年の夏は異常な冷夏に見舞われ、6月、青州の胊山で霜害が発生し[5]、さらに秋7月、青州で雪が降り、穀物収穫に被害を及ぼした[6]。青州といえば、現在の山東半島の北緯37°の場点にあり、日本のいわき市、北米のカリフォルニア、中東のシリアなどと同緯度に当たる。茨城県で8月に雪が降るようなものである。

「There seems to be something wrong」, I assume Chinese people of Lyan Dynasty must had begun to think. But the time was too late, the chains of the extravagant disasters already started. In the summer of the 537, Lyan people suffered from the extraordinary cold summer, and in the same year, in June in Qingzhou, people suffered from the severe frost damage, ( Nanshi 1.7.4.3 ) and in the same year, in January, in the same Quingzhou snow fell thus it had enormous impact on the production of crops. ( Nanshi 1.7.4.3 ). If you account for Qingzhou province, it is situated at the same latitude of 37 degree in the northern hemisphere, same with Japanese Iwaki City, and the U.S's California, ans also with that of Syria. It would be equivalent with that snow falls in August at all these areas at the same time.

当然のようにこの後538年飢饉が発生し[7]、政府の当局では八月甲辰、南兗州など十二州に詔を下し、飢饉が発生した地域では、租税を滞納している者に特赦を与え、今年の三調の徴収を禁止[8]するほどの対応をとらなくてはならなかった。異常気象は続き、皇帝は541年11月、「『天の道を用いて、地の利を分かつ』とは古代の聖人の訓辞である。没収された耕作地や桑畑・住居について、皇室領となった土地を除いて、全て貧民に分け与えよう。彼らの労働力を調査し、耕作地を授けることとせよ。聞くところによると、近頃では有力者や富豪が多くの公田を占有し、高額の小作料を徴収して、貧民に与えるなどと、今日の行政に甚だしい損害を与えている。今より一切、公田を有力者に貸し与えることを禁ずる。すでに貸し与えられた分については特別に追及しないこととする。富豪が貧民に種籾を与え、ともに田地を経営するのであれば、この禁令の例外とする[9]」という詔を発するに至る。

Gradually updated due to the enormous difficulties of the translation.

Conclusion

Nanshi, the chronicle of events of the Lyan Dynasty, records multiple events of unusual climate events that occurred from the 534AD to 550AD, as also the 北史 ( I am sorry I can not translate ), the chronicle of events of the Northern Wei Dynasty records the same very unusual climate events which occurred around the same period that were recorded by Nanshi.

The main descriptions of the abnormal weather is found as mentioned in the other answers from the 北史/南史 beishi/nanshi which mentions snow in September and August respectively.

As for the highlighted quote I've managed to track it down to a passage in the 资治通鉴第一百五十七卷 Zizhi Tongjian 157th Juan (13th chapter in the Liang Dynasty sections). The original texts has the following:

是岁，魏关中大饥，人相食，死者什七八。

which translates roughly to:

AD536, Large famine in Wei, Cannibalism reported, death seven to eight in ten.