I can confirm this adapter works with the IIgs and with my 15.75khz RGB capable CRTs, I don't have any LCDs here that I can test with that ability.
http://www.belkin.com/support/dl/f3h1381revised.pdf
I had to fiddle with switches 5, 6 and 7 if I recall. This adapter appears to have a sync splitter inside of it to split composite sync to H+V sync. the rotary knob and the rest of the switches should have no effect on the IIgs as it doesn't use the sense pins like a Mac does. I have the knob at 0 and switches 1-4 off, 8 on.
Pinout for the IIgs port is here: https://support.apple.com/kb/TA36688?locale=en_US
Pinout for Centris/Quadra is here: https://support.apple.com/kb/TA47448?locale=en_US&viewlocale=en_US
Looking at that, your Mac to VGA adapter may not connect IIgs pin 3 to anything, you may have to wire IIgs pin 3 to VGA pin 13. Most VGA monitors will take composite sync on their H sync line. If not, wire IIgs pin 3 to VGA pin 14 as well.. If that STILL doesn't work, the last ditch effort is a sync splitter. Amiga users don't have to worry about this as that machine outputs composite along with seperate horizontal and vertical sync signals.
http://www.belkin.com/support/dl/f3h1381revised.pdf
I had to fiddle with switches 5, 6 and 7 if I recall. This adapter appears to have a sync splitter inside of it to split composite sync to H+V sync. the rotary knob and the rest of the switches should have no effect on the IIgs as it doesn't use the sense pins like a Mac does. I have the knob at 0 and switches 1-4 off, 8 on.
Pinout for the IIgs port is here: https://support.apple.com/kb/TA36688?locale=en_US
Pinout for Centris/Quadra is here: https://support.apple.com/kb/TA47448?locale=en_US&viewlocale=en_US
Looking at that, your Mac to VGA adapter may not connect IIgs pin 3 to anything, you may have to wire IIgs pin 3 to VGA pin 13. Most VGA monitors will take composite sync on their H sync line. If not, wire IIgs pin 3 to VGA pin 14 as well.. If that STILL doesn't work, the last ditch effort is a sync splitter. Amiga users don't have to worry about this as that machine outputs composite along with seperate horizontal and vertical sync signals.