Yes you can put toothpaste on a mosquito bite to help it go away sooner and help ease the itching of the mosquito bite.
The best kind of toothpaste to use for mosquito bites is toothpaste with menthol as the menthol helps to soothe the mosquito bite.
The menthol toothpaste acts as a cooling agent which soothes the itch of the mosquito bite so you do not want to scratch it.
You can also help soothe a mosquito bite by heating up a spoon and applying to the mosquito bite or applying a cool compress.
Using Calamine Lotion on the mosquito bite or even some ice with an ice pack or applying some over the counter hydrocortisone cream.
Most mosquito bites do go away on their own as well even without treatment and the mosquito bite should go away within a few weeks to a month or two at most.