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When civil registration was introduced in England and Wales on 1 July 1837, Quakers were included in it, but had special arrangements. However, those arrangements ensured that their marriages were reported to the GRO and so should appear in the indexes. Like the Church of England (and Jews), and unlike any other nonconformists at that stage, they were allowed to celebrate and register marriages without the local registrar being present. Each Monthly Meeting was issued with duplicate register books which look just like those issued to C of E parishes (apart from a slight difference of wording), and appointed a Registering Officer to record marriages. Like a Vicar, he had to make a quarterly return to the Superintendent Registrar (including a nil return if necessary, as it often was). Quaker marriages at this stage were only valid if they took place in a Meeting House and both parties were Quakers; later (in two stages in 1860 and 1872) the law and Friends' regulations were relaxed to allow marriages involving non-members.
It follows that all Quaker marriages in England and Wales from July 1837 ought to appear in the GRO indexes. There are known problems of occasional mistakes and omissions in the indexes, but these do not only affect Quakers. Many, but not all, Monthly Meetings have deposited their early post 1837 registers with their other records in local record offices, and this may be a source you can check.
Friends continued after 1837 to draw up lengthy marriage certificates in the traditional form, signed by all present (and still do). But these were in addition to, not instead of, the statutory certificate in the usual civil registration form.