Hello again,
This case is a bit more complicated just because it is Exeter. The quarter sessions records which are indexed are annual returns made by the petty sessions of affiliation orders so they are different from the disputed paternity cases which you mention. At this date, it was the responsibility of the mother to apply for maintenance for her baby, not the overseers of the poor. It is possible that William, as an illegitimate child, may have been born in the workhouse, but Exeter workhouse records were destroyed in the bombing in 1942. Exeter didn't have a poor law union, but a Corporation of the Poor, which dealt with all the poor law cases in the city parishes.
You could guess that William was born in the workhouse, and in that case he may have been baptised in Heavitree parish church, but it would be quicker to ask the Superintendant Registrar at Exeter to search for a birth record for you.
Sorry to ramble on

Sue