This is a summary of what I have been able to find thanks to your help:

James Quayle & Joney (?) married date unknown
William 11 Feb 1808 Lezayre
James 23 Feb 1812
Philip 18 Mar 1814

Philip Quayle
m. (1) Margaret Bridson (abt 1806) 8 Aug 1840 Malew

Eleanor abt 1840 Lezayre

James Henry 28 Jan 1844

John Thomas 3 Jan 1847
The IGI records Ellen (Eleanor) as marrying Thomas Holmes in 1862 (submitted by LDS member)

m. [2] Catherine Cowin (b. 1820, father David Cowin) 4 Sep 1850 Lezayre

Ann Jane 22 Jun 1851 Lezayre

Margaret 10 Oct 1852

Elizabeth 17 Nov 1853

Thomas 25 Mar 1855 - 6 Jun 1855

William 30 Mar 1856

Robert 26 Jul 1857

Esther 8 Aug 1858

Emily Anne 4 Mar 1860

Eliza Jane 20 Jan 1861

Edward Philip 11 May 1862

1871 Lezayre, Church Town
Philip Quayle (married) 56 Ag. Labourer Lezayre
Elizabeth 16 Domestic Servant ''
Esther 12 Scholar ''
Emily 10 '' ''
Eliza J. 9 '' ''

1871 Braddan, Ghofellan General Lunatic Asylum
Name: Catherine Quayle
Age in 1871: 52
Estimated birth year: 1819
Relation: Inmate [Lunatic]; wife of Labourer; born Isle of Man

[maybe as a result of all those children?]

These may be William and Robert in 1871
Lezayre, Glenduff House (farm of John Mylichurst [sic] 51 Maughold)
William Quayle 15 Lezayre, Isle of Man, Isle of Man Servant Lezayre Isle of Man

Lezayre, Cranniag (farm of John Cowly, 69, Lezayre)
Robert Quayle 13 Lezayre, Isle of Man, Isle of Man Servant Lezayre Isle of Man

For Margaret Bridson, there are three candidates

1] Christening: 13 Jul 1806 Malew, Isle Of Man, England. Parents John Bridson : Isabel McGray

2] Christening: 10 May 1807 Malew, Isle Of Man, England. Parents Paul Bridson : Isabel Moor [alternatively 10 Apr 1807, Isabelle Moore]
3] Christening: 06 Dec 1807 Malew, Isle Of Man, England. Parents David Bridson : Margaret Christian

You have rightly identified Jane Gawne as the daughter of Thomas Gawne and Catherine (Curphey) of Andreas.
Jane married John Thomas Quayle on 3 Dec 1870 and had a daughter, Norah Jane, christened 22 Jan 1871. John died sometime before 1881 and Jane went to London where she married Charles Forbes Kidd in Lambeth, Surrey on 18 Oct 1881. Their third son William, born 1886, was my grandfather.
Norah stayed on the island and was raised by her Gawne grandparents (in 1891 she appears as Norah Gawne) but by 1901, she too had gone to the mainland and married Joseph William STUBBS of Maryport, Cumberland. They married in 1894 ( Norah under her name of Quayle) and by 1901 had four daughters: Mary (1894), Lilian (1896), Betsy (1899) and Phoebe Catherine (1901).

I cannot see any connection to Carol's family of Quayles so far but, as yet, neither James Quayle nor his wife Joney have been identified, neither has their marriage been found.
Maybe somebody will recognise someone here.
Thanks again