Really depends what mount you have for your prior GMRS antennas. NMO is a pretty common one, but the 770 is a UHF base. uhf to NMO adapters are pretty easy to find on amazon in the $10-15 range.

can you swap the spring from one of the other ones onto the nagoya? i won't rule it out, but it's unlikely, and would almost certainly mess with the tuning.

the 1/4 wave antenna suggestion is a good one, providing you have a good ground plane for it. failing that, i'd lean toward something extra flexible, like a signal stalk and grab the swr meter or VNA and start testing, probably give it a slight trim to bring it in line for gmrs. either will be less likely to snag overhead obstacles than the nagoya.