This is actually complex. But QE, in most cases but not all (for example, QE3) involved the purchase of assets from the non-bank sector, not purchases from banks (in the US, banks do not hold a large amount of US govt securities).
When QE did involve the purchase of assets from banks (i.e. QE3 with MBS/ABS purchases, and in the EU where banks held a lot of govt debt) then it did lead to increase lending but the effect varied significantly (for example, in the EU the effect was tiny because most banks in the region are functionally insolvent).
So QE is not thought to have any particular effect on bank lending, and is generally not used with that aim in mind now. I think this was thought to be true in 2008 but we know now that it has no/little effect (i.e. QE is mainly effective through portfolio rebalancing). And central banks that did try to increase bank lending (the UK in the mid-2010s, and the ECB...always) use other techniques (i.e. FLS, LTRO...which, iirc, are direct lending to commercial banks by central banks...I think, I don't follow this stuff closely anymore).
Just generally too, if commercial banks are holding a lot of govt debt then that is usually a sign of problems elsewhere (i.e. the EU).