Gabby was always an oddity to the others in Griffinstone. Her perky demeanor, friendliness, and determined optimism put her at odds with a lot of the griffons there. Like many others around her age, Gabby was raised by a single brooder, Grandpa Gruff, instead of her bio parents (A common practice among griffon society)*. Gruff would never admit it, but he had a soft spot for Gabby - the girl never seemed to lose her kindess and compassion, even though all the others grew out of it. That, and she was the runt of her brood, shorter and stouter than her peers.  
As a young teen, Gabby feels a need to interact more with her neighbors and do something to help, so she takes up mail-carrying. It was a little difficult, with the griffons not really having stable places to live or even mailboxes most of the time, but she got the job done. Then she notices Gilda acting more friendly than usual. From Gilda she hears about the ponies of Equestria, and the importance they put on friendship with each other. The grey griffin new exactly where she wanted to go. After some misadventures with the CMC, Gabby kept in touch with them as she took her friendship lessons back with her.
 

 
She actually moved herself out of Griffinstone after she got older; sure it was where she grew up, but Equestria just felt more like home to her. She moved close to Ponyville, and took up several different jobs as a jack-of-all-trades, helping out whoever needed it. After a few years, she took up a job writing an advice column in the Ponyville newspaper under the ‘penname’ Gabby Gums.
 
Applebloom especially close with Gabby, and was the one who introduced her to Spike. The two got along quite well, having common interests in comic books and bonded over being the odd one out in their respective species. The two developed a mutal crush as they spent more time together, but were doubtful that their feelings were returned. ie,
 

 

 
 
*Also the Grandpa Gruff headcanon is a thing I, borrowed from Loryska and one of her world-building posts!: World Bits: The Griffin