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Do we have any definite numbers on the sizes of Sector Battlefleets?

Yes

Each battlefleet consists of 50 to 75 warships of varying size, although in some sectors this will more or less, according to the importance of the sector and the number of enemies it must contend with. As well as these destroyers, frigates, cruisers, and battleships...

p. 86, Battlefleet Gothic rulebook

It is 50 to 75 warships of all sizes, meaning not all are capital ships and some are escort sized.

Sectors near the eye of terror or Terra have more while "non"-essential sector's have less (one of the reasons the gothic sector was so quickly over run)

While that is all good and true, I'd also like to point out that fleets grow and shrink by the needs of the authors all the time, so this number is not necessarily reliable.

The "correct" answer is "how big do you want it to be?"

A sector's battlefleet is normally dispersed on patrol and on "showing the flag" missions to the worlds of the sector, while the heavier ships tend to stay near the main naval bases. For the most part, an Imperial Navy ship is enough to see off most pirate ships, which are about escort sized. The BFG rulebook gives the standard sector as a 200 light year cube, in which the worlds of the Imperium normally cluster in sub-sectors, like oases in the desert, separated by tracts of unknown (aka wilderness space).

In the event of major incursion, such as the Gothic War, then the sector fleet may find itself overwhelmed and forced to withdraw to bases while awaiting reinforcements from nearby sectors in order to counterattack.