Is it permissible for a mother to secretly give her saved money as zakat, without her husband's knowledge, after he had forgiven her for concealing the money?

If the money is fully owned by your mother, has reached the nisab (minimum threshold), and a full year (hawl) has passed on it, then she must pay zakat (2.5% of the money). Your father cannot prevent her from doing so, and she is not obliged to inform him of it, as his knowledge is not a condition for zakat to be obligatory or for its payment to be valid.