No — you owe nothing for the years before you accepted Islam. The Prophet (peace be upon him) told 'Amr ibn al-'As that Islam wipes out whatever came before it, and Allah says of the disbelievers that if they desist, what has already passed will be forgiven. Your worship begins from the day you enter Islam; you are not required to repay a lifetime of missed salah, Ramadan, or zakat. Simply start now and keep going.
Informational, not a personal fatwa. Consult a qualified scholar for rulings on your situation.