If i moved in with someone in the future, what is the best month to do so for tax purposes?

Depending on your situation, there may or may not be any benefit to being common law. For example, if you have one partner with a high income and one with a low income, then the high earner might be able to claim the Spouse or common-law partner amount and lower their taxes owing. But if both partners have high incomes, then then living common law may not really change their income tax situation much.

If you don't have a child with your partner, you would have to live together for 12 months before you would be considered common-law So, if you do stand to benefit, then you would want to move in with each other as soon as possible.