‘At the time of capture the drake had been trying to swallow a frog, by quapping with its bill, which held one of the legs.’

Lovely word this, though not for the frog (or the drake, who was quapped in turn).  Another fine sounding verb which seems to be surviving only at a regional level.

As used here, I expect it draws on the OED meaning of ‘to tremble, wriggle, writhe’, as you can imagine the duck gulping to try to prevent the frog’s escape from the hatch.

The more general meaning, now obsolescent, is of throbbing (including pain) or pulsating, palpitating, and similar.  So you can confidently refer to your quapping / quopping heart.

A poetic use describes the heaving or undulating of the sea.

Other forms include: quape, qwape, qwappe, quappe, quapp.

Source: Henry Williamson, Tarka the Otter: His joyful water-life and death in the two rivers, illus. C.F. Tunnicliffe (Harmondsworth: Puffin Books, 1976 (1927)), p. 51