Can your teeth become loose after menopause?

No. After menopause teeth does not become loose, unless you have disease of the gums or other dental problem.
Maybe. Menopause is a complex series of hormonal changes, and it is common for women in these years to lose bone density. If superimposed on this process there is periodontal disease you will likely have a fairly rapid change in the mobility of your teeth. Occlusion must be checked and equilibrated, by doing so spread forces amongst more teeth.