The office responsible for marriage is called the Standesamt. You should contact the Standesamt in Lüübeck and ask for an appointment. Here is the information about it (via online translation):
They will inform you which documents they require. In general these are a birth certificate, passport and certificate of eligibility to marry (Ehefähigkeitsbescheinigung). The last document is usually issued by the Georgian embassy.
It is not clear to me what kind of residence paper you have and whether you are still in the asylum process with the paper 'Aufenthaltsgestattung' or you have received a residence permit 'Aufenthaltstitel'. In either case, as refugees you cannot contact the embassy of your home country for the certificate you need. For such cases, the Standesamt can accept an affadvit about your marital status given to them.