I am not a FL attorney, laws vary from state to state, therefore you should always consult a local attorney.
Your grandmother could deed the property to someone else and have that recorded immediately. As a general rule, states follow a "first in time, first in line" rule so that the deed that is filed first will take priority over a previous unrecorded deed.
Note however that this situation sounds like you will end up in court at some point in time. So, should your grandmother decide to deed the property to your mother, she should do so in front of witnesses who can later testify that she did so freely and voluntarily and that she was not under any duress or coercion and was of sound mind at the time. As a general rule, it is better if the witnesses are not immediate family members who have an interest in the estate. There is no guarantee that this means you mother will win - because she is taking with notice of the prior transfer. That is why, as you indicate above, you are going to be advised to see a local attorney.
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A quit claim deed is only valid if it is delivered to the grantee. If a deed was executed and not recorded until more than three years later, there is a question as to whether or not there was an adequate delivery. In addition, if your grandmother was not competent or was acting under undue influence, it could be set aside. As you surmised, this is a question that cannot be completely answered online. You should consult an experienced real estate lawyer in your area about the best way to proceed in your case.
Disclaimer: This answer is provided for informational purposes only, does not constitute legal advice, and does not create an attorney-client relationship. Actual legal advice can only be provided after completing a comprehensive consultation in which all of the relevant facts are discussed and reviewed.
Florida is what is known as a "notice" state. Generally, that means that (as another attorney mentioned) if your grandma issues another quit claim deed to your mother and your mother records her deed, then she most likely will own the house. Are you certain that a deed was delivered to the aunt and accepted by her? If not, then the home still belongs to your grandma to do with what she will. Can you "stop" the quit claim deed? No. If it was drafted properly, delivered to the aunt, was not the result of undue influence, and was accepted then the home belongs to the aunt, regardless of whether or not the deed has been recorded. But, all those factors must align. Is it possible to ask your grandma about the transaction? If not, or if she is no longer competent, how would she convey the property to your mother?
Carol Johnson Law Firm, P.A. : (727) 647-6645 : firstname.lastname@example.org : Wills, Trusts, Real Property, Probate, Special Needs: Information provided here is anecdotal and should not be relied upon or considered legal advice. Every matter is different and answers given here are general in nature and may not reflect current Florida law at the time you are reading this posting. Please contact me if you feel you need additional assistance with your matter.
As others here have pointed out, if your mother is first to record her QCD, then perhaps she will prevail. Your grandmother's state of mind when she wrote the QCDs may also be an issue. Additionally, check your grandmother's will; it may contradict or interfere with the direct transfer of title contemplated in either QCD. Finally, talk with your grandmother. As I am also a mediator, I always pose this (obvious) approach to settling disputes, especially between family members.