Mr Abbot was appointed early last year and moved from Cairns, Australia, to start the job last July.
He has been in Australia, with his family, since just before Christmas, but has so far been unable to return.
His family had been due to join him late last year, but, similarly, had not been able to due to the "significantly obstructed borders".
DQ chairman Richard Thomas said Mr Abbot had to make the "difficult decision" due to the "incredibly challenging and ongoing conditions brought about by Covid-19, border restrictions, and the need to put his family first".
"The DQ board is very sad to see Paul go, but can empathise with his extremely difficult position having his family stuck on the other side of the Tasman."
Mr Abbot will finish next month.
