We need to look at the circle of fifths to answer your question correctly. The range of notes we'll look at are :
... Db Ab Eb Bb F C G D A E B F# C# G# ...
You might even see the beginnings of a pattern... the notes start as F C G D A E B, then start over again sharped, ie F# C# G# D# etc. Likewise, if we start at B and move the other direction, we see B E A D G C F then Bb Eb Ab Db etc.
So the order of sharps is F C G D A E B, and the order of flats is reversed, ie B E A D G C F. Memorize this, its important!
We always start at *C* - its the natural key, it has no sharps or flats. If we want to move to the key of G, we are moving one note fifth-wise, ie one note "sharp-wise". We change one sharp, and that note is F -> F#.
If we want to move to the key of F, we are moving one note "fourth-wise" ie towards the flats, so we add one flat, which means we change the B -> Bb.
So how do we get to the key of G#? We have to move from C to G to D etc... its a movement of *eight* sharps, which means we would end up with eight sharps in the key signature, the notes would look like this -
G# A# B# C# D# E# F## G#
This is completely unusable. One of the maxims is to use whatever key signature has the fewer number of sharps or flats. So what is this key signature in flats? G# = Ab, which is four notes from C in the "fourth-wise" direction, which would give us these notes -
Ab Bb C Db Eb F G Ab
Count out the intervals ( W W H W W H ) to double-check.
So we would not use G#, we would use Ab instead.
Note that you said "G#". We assume major unless you specify minor! If you meant G# minor, then that changes things.... when figuring out the key signature of a minor key, we look at its relative major, ie, the note one and a half-steps up.
So just like Am is the relative minor of C (or C is the relative major of Am), B is the relative major of G#m. The key of B is five sharps, and would look like this
B C# D# E F# G# A# B
And the G#m would then look like this -
G# A# B C# D# E F# G#
Again, count out the intervals to double-check.
And there you go! Google "chord transposition", "circle of fifths", and probably "major key modulation" to learn more.
Saul