It does not appear that John Palmer, son of George & Elizabeth Palmer, was married or in the least, he left no issue. He died in 1710 leaving his Will #161 ½ , dated 3 Apr 1710 and proved 25 Apr 1710, PA Will Bk C, pg 203.
He styled himself as of "the City of Philadelphia. carter"; he named his mother Elizabeth Fitzwater, brothers Thomas Palmer and William Palmer and cousin (nephew) John Ricketts; with brother William Palmer as Executor. [PA Will Bk C, pg 203)
[John Rickets was the son of John's sister Elizabeth Palmer and her 1st husband Isaac Rickets, m. 1687. After Isaac's death in 1692, she married 2nd Ralph Jackson in 1695 and had 2 more children, Charles & Elizabeth Jackson. Elizabeth (Palmer) Ricketts-Jackson died in 1704 and her son John Ricketts died in 1712, leaving a Will dtd 9mo/Nov 6, 1712, proved X'ber/Dec 20, 1712, PA Will Bk C, pg 327. He stated his desire to be buried at the burying ground at Philadelphia next to his mother; named (half) sister Elizabeth Jackson, uncles Thomas and William Palmer, William as Executor and one of the witnesses was his step-father Ralph Jackson]
I have attached a copy of the Will of John Palmer herein.
He should not be confused with John Palmer of Concord, wife Mary and children John Palmer Jr, Catherine Skears, Alice Buffington, Margery Kerlin, Ann Trimble and Mary Trimble; Executors: wife Mary and son John. His Will was dated 22 Feb 1739/40 and proved 16 Jul 1742, Chester Co Will Book B, pg 110.
Hope this helps your research!