It would be better having a nikah with all conditions! If according to the school of fiqh both of you are following the agreement of your (I suppose you are female) guardian is not necessary you are already married with no doubts about the validity of marriage, if not you should consider telling your parents and have a correct nikah.
Be aware that you should never fear being truthful and correcting what was wrong. Most schools of fiqh consider the agreement of the guardian as essential an imam of a mosque can only replace him under certain conditions, which today are hardly applicable, but a woman should always seek the agreement of her guardian first. Also be aware that even if you married without the agreement of your guardian a shari'a court would hardly consider separating you if for example you were already having kids or being pregnant.
And finally a nikah certificate is not a requirement for a marriage being valid in Islam as it is an innovation of our modern times which has clear benefits and therefore shouldn't be regarded as bid'ah. However it might be a requirement for a civil notification or record of a marriage which declare this marriage as legal according the local law.