Is it permissible to continue placing Google ads on a Quran website, while being careful to filter out impermissible ads, and to use the profits to support the website, despite objections to them?

There is no harm in including advertisements on your website, provided they are free from শরعي (Islamic legal) prohibitions, as Allah Almighty says: "And do not cooperate in sin and aggression." You must also verify whether Google ads promote Christianization or immorality to avoid them.