Recently, in D&D 5e game, I got a young dragon and I want it to become at least mature (if ancient is not a possibility). However, I wouldn't like to wait for too much. Is there some kind of potion/spell that can speed this process up or some other way?

The campaign won't last a year, so my dragon cannot wait 100 years to mature.

Depends on what you call "mature".

Raw aging can be done via magical means, as accurately described in K-T's answer.

But an adult dragon has a better stats block than a young dragon not only because of it's age, but also because of it's experience (and growth, which may be treated separately from aging by your DM - as concluded here)

Your DM will decide what stat block he wants to use for a "magically aged young dragon" - and how the creature behaves, depending on the chosen method.

By the way, he will also decide if a tamed adult dragon fits his campaign or not - which may render your project from easy to painful and impossible.

Lots of Potions of Longevity

Potion of Longevity

Potion, very rare

When you drink this potion, your physical age is reduced by 1d6 + 6
years, to a minimum of 13 years. Each time you subsequently drink a
potion of longevity, there is 10 percent cumulative chance that you
instead age by 1d6 + 6 years. Suspended in this amber liquid are a
scorpion's tail, an adder's fang, a dead spider, and a tiny heart
that, against all reason, is still beating. These ingredients vanish
when the potion is opened.

The first few times the dragon drinks the potion, it'll probably get younger. But after the 10th potion, the dragon will be guaranteed to get older every time. Getting enough Potions of Longevity might be tough though, due to them being very rare.

Alternatively...

Get a ghost to scare your dragon

Consider the ghost's Horrifying Visage action:

Horrifying Visage. Each non-undead creature within 60 ft. of the ghost
that can see it must succeed on a DC 13 Wisdom saving throw or be
frightened for 1 minute. If the save fails by 5 or more, the target
also ages 1d4 x 10 years. A frightened target can repeat the saving
throw at the end of each of its turns, ending the Frightened condition
on itself on a success. If a target's saving throw is successful or
the effect ends for it, the target is immune to this ghost's
Horrifying Visage for the next 24 hours. The aging effect can be
reversed with a Greater Restoration spell, but only within 24 hours of
it occurring.

If the ghost uses this action, and the dragon's Wisdom saving throw is 8 or below, it will age an average of 25 years. A ghost can try to scare the dragon once every 24 hours. That might not be very quick compared to drinking a bunch of potions, but it's a lot cheaper!

Honestly, there's not much time or age manipulation available in DnD 5e.

Probably because time is one of the biggest headaches for any GM of any game to deal with, especially on the small scale, where you could use even a half-second of time manipulation to reroll a bad roll, undo an enemy's action, cast multiple spells in one turn, or any number of other crazy things. A Time Turner in DnD would be a Deity-level artifact. I'd never put one in my game, tbh.

But, there are some options not already covered:

• You could cast Wish and use it to age your dragon. This is subject to the risk the DM will mess with you for using Wish outside its standard parameters.




• Polymorph might work, DM willing.




• Find a plane with much faster time than the material plane. Maybe
  make a deal with an Archfey to bring your dragon into his realm for a
  century or two of his time, while only a few weeks pass for you.

Honestly, this would probably be my favorite option as a DM because
it is story driven and not just a mechanical trick requiring
adjudication.

But why do this?

You might want to ask why you want an Adult dragon pet. It'll have a solid chance to be smarter than you depending on stats, much stronger, and entirely free-willed with no reason to obey or serve you. Even the Good-aligned dragons have a towering ego and lots of pride. It may not go as well for you as you think, and no GM is going to give players an overwhelming power advantage that will let them steamroll the game. An Adult dragon gets Legendary Actions and Legendary Resistance, and allowing Players to control those on any kind of regular basis is a huge balance problem.