Can you make berries grow faster?

You could with a mod, but these are hard to come by.
Just keep watering them every so hours and they'll grow in no time.

Water them.

If you save then turn off your game for about half an hour and then turn it back on they should be grown.

No.

Yes water them you also get more berries this way

I planted a berry in route 120 then I was shocked when I saw the seed sprouted in an instant.. I have searched every forums and I can't find an answer.. Can someone tell me how can this occur again?

The instant sprout happened to me too. I planted a Leppa Berry in route 123 by the berry master's house. It produced 2 fruits instantly, after some blue stars circled the soil.

They spouted fast because of the rain

I've tried this too.
In my opinion, have a Groundon in your team at the first place.
Go to route 120, cross the bridge and walk south.
Walk to the right until you see a small area with a small tree
Use cut there, after that, plant the berry in the soil there.
(Since Groundon's drought intensifies sun rays and it is raining too, the berry will sprout more quickly.)
Water the berry after every 5-10 mins and you'll have your berry in no time

I planted a berry near the daycare's house, and I just wait then there the berry is ready to harvest.

Name Sapling Gr. Taller Flowered Pick Berry
---------- -------- ---------- -------- -----------
01. Cheri 3 hours 6 hours 9 hours 12 hours
02. Chesto 3 hours 6 hours 9 hours 12 hours
03. Pecha 3 hours 6 hours 9 hours 12 hours
04. Rawst 3 hours 6 hours 9 hours 12 hours
05. Aspear 3 hours 6 hours 9 hours 12 hours
06. Leppa 4 hours 8 hours 12 hours 16 hours
07. Oran 3 hours 6 hours 9 hours 12 hours
08. Persim 3 hours 6 hours 9 hours 12 hours
09. Lum 12 hours 24 hours 36 hours 48 hours
10. Sitrus 6 hours 12 hours 18 hours 24 hours
11. Figy 6 hours 12 hours 18 hours 24 hours
12. Wiki 6 hours 12 hours 18 hours 24 hours
13. Mago 6 hours 12 hours 18 hours 24 hours
14. Aguav 6 hours 12 hours 18 hours 24 hours
15. Iapapa 6 hours 12 hours 18 hours 24 hours
16. Razz 1 hours 2 hours 3 hours 4 hours
17. Bluk 1 hours 2 hours 3 hours 4 hours
18. Nanab 1 hours 2 hours 3 hours 4 hours
19. Wepear 1 hours 2 hours 3 hours 4 hours
20. Pinap 1 hours 2 hours 3 hours 4 hours
21. Pomeg 3 hours 6 hours 9 hours 12 hours
22. Kelpsy 3 hours 6 hours 9 hours 12 hours
23. Qualot 3 hours 6 hours 9 hours 12 hours
24. Hondew 3 hours 6 hours 9 hours 12 hours
25. Grepa 3 hours 6 hours 9 hours 12 hours
26. Tamato 6 hours 12 hours 18 hours 24 hours
27. Cornn 6 hours 12 hours 18 hours 24 hours
28. Magost 6 hours 12 hours 18 hours 24 hours
29. Rabuta 6 hours 12 hours 18 hours 24 hours
30. Nomel 6 hours 12 hours 18 hours 24 hours
31. Spelon 18 hours 36 hours 54 hours 72 hours
32. Pamtre 18 hours 36 hours 54 hours 72 hours
33. Watmel 18 hours 36 hours 54 hours 72 hours
34. Durin 18 hours 36 hours 54 hours 72 hours
35. Belue 18 hours 36 hours 54 hours 72 hours
36. Liechi 24 hours 48 hours 72 hours 96 hours
37. Ganlon 24 hours 48 hours 72 hours 96 hours
38. Salac 24 hours 48 hours 72 hours 96 hours
39. Petaya 24 hours 48 hours 72 hours 96 hours
40. Apicot 24 hours 48 hours 72 hours 96 hours
41. Lansat 24 hours 48 hours 72 hours 96 hours
42. Starf 24 hours 48 hours 72 hours 96 hours
43. Enigma 24 hours 48 hours 72 hours 96 hours
...
Just be patient n keep waiting..after watering that seeds..

After you watering that seeds.. U save firstly.. N change the time..