Is it possible that elimination of one mosquito species could lead to an increase in other mosquito species in the area?

Yes. The issue of competitive replacement, also called niche replacement, is a possibility that the World Health Organization and others have recommended should be considered in risk assessment. There are two parts to the question of whether this could lead to harm, however. The first part of the question is whether it might happen. The second part is whether it would result in increased disease transmission. For example, there is evidence for competitive replacement of Aedes aegypti with Aedes albopictus, where they have overlapping distribution. But Aedes albopictus is widely thought to be less competent than Aedes aegypti for transmitting arboviruses such as dengue so it is unlikely that this would result in substantially greater disease risk overall. An extensive study of the effects of insecticide-based vector control programs targeting Anopheles species in Africa suggests that reduction in numbers of Anopheles gambiae mosquitoes was sometimes followed by a local increase in other related species, but these other species were less efficient vectors of malaria transmission.

For more information:

https://www.who.int/publications/i/item/9789240025233

https://parasitesandvectors.biomedcentral.com/articles/10.1186/s13071-021-04975-0

https://www.youtube.com/playlist?list=PLbopRNGowKJ_z1k9Sqxt26ONibFCjlgKe

https://malariajournal.biomedcentral.com/articles/10.1186/s12936-021-03674-6