During the Spanish Reconquista, did much of the Muslim population convert to Catholicism?

Different states treated the Muslims differently.

Castile and it's possessions were known to be extremely aggressive towards Muslims and Jews. However, for most of the period of the Reconquista, this was in the form of a heavy tax burden placed on non-Christians, similar to the Jizya (the Muslim tax on non-Muslims) in place earlier, although probably much heavier. However, prior to the completion of the campaigns, this monetary contribution was probably more beneficial, as it encouraged conversions in the same way the Jizya did; that is, without having to resort to costly and dangerous population purges. The key thing is that any attempt to avoid the heavy (often crushing) burdens placed on them was harshly punished. This was an easy way to target individuals. In addition, huge swaths of land were appropriated by incoming Castilian nobility, which was often still worked by the formerly Muslim peasantry. This gave landowners another important leverage over those that worked their land.

In Aragon, things worked a little differently. Initially attempts were made to incorporate much of the existing leadership structure of the conquered areas of Catalonia and Valencia, but a revolt in the 13th century by Muslim leadership put the kibosh to that. However, within the coastal cities relatively lax treatment was available to Muslims and Jews, as these populations and their trading relationships were seen as very valuable.

In both Castile and Aragon, things changed drastically in the very end of the 15th and early 16th centuries. Muslims and Jews were given the option of conversion of expulsion, and this is when the Spanish Inquisition earned it's reputation. Paranoia about falsely converted Muslims and Jews was what gained them that.

I am not as familiar with Portugal however. It does seem that in the early 16th century they too took a harsher tone against Muslims by expelling all Moors, but I am not as certain as to the levels of persecution prior to that. It is important to note that Portugal completed their own Reconquista far before Castile.

As for the ultimate fate of the Muslim population of Spain, most of them would be descendents of the original inhabitants there before the invasion by the Caliphate who had converted over the centuries, and probably not all that ethnically different from their northern neighbors. The Muslims in leadership positions were the ones who would be the ones most likely to be expelled on sight or killed. Southern Spain has a population with higher 'Moorish' blood origins, likely thanks to greater intermingling of the population with North Africa's due to proximity as well as time spent under their rule (leading to the distinct 'Andalusian' ethnic/cultural group.