No, you can't have a second civil marriage in Canada. But you can have a (legally meaningless) religious marriage if your religious organization will perform one without the civil part - you must be specific with them about this, you only want the religious ceremony (because already legally married).
Background: in Canada marriages are mostly performed by individuals (eg religious leaders) as agents of the government (sort of like being deputized). They can perform whatever ceremony they like (or that the church or religion allows), and they carry out the civil functions (registering marriage, witnesses signing docs) as part fo the process (usually quietly and separately but depends).
Large organized churches (eg Catholics, Anglicans, etc) have specific rules on what their ministers are allowed to do (what types of ceremonies, where, minimum requirements, etc) and call weddings or marriages. If they will not /cannot perform just a religious marriage without the civil part, they may offer something like a 'blessing of union' or some other name.
In other words, ask your religious leader/community what is possible.