Can i say 'çabukça'
But of course! There are a few other derivation of çabuk (namely either that word or çabucak) that you can say as well! :)
Why is this sentence wrong''calm down, you need not to come quickly''?
Doesn't sound right to me also
Did they mean
'you don't need to come quickly'
"Relax, it it not necesary to come quickly" should be accepted.