You would not be able to be married in an Orthodox Jewish synagogue nor mostly likely in a Conservative Jewish temple. From an orthodox perspective intermarriage is the greatest threat to Jewish continuity and is greatly discouraged. You might want to investigate Judaism to see if it appeals to you, and if you would consider converting even if you were not marrying a Jewish woman. In that case after a period of study and Jewish practice you could convert and be welcomed even in an orthodox synagogue.
How wonderful that you have found someone that you plan to share your future with.
I understand from your question that you have not converted to Judaism but that your girlfriend would like to get married in a synagogue (I assume in a Jewish wedding ceremony).
Jewish law (halakha) requires that both participants are Jewish by birth or have converted to Judaism prior to marriage. The traditional branches of Judaism (which includes Conservative Judaism) will not allow you to get married in a synagogue or elsewhere in a Jewish wedding ceremony and neither will rabbis in those branches officiate at such a marriage.
The question is really whether you want to have a Jewish wedding and be married by a rabbi? If the answer is yes, and if you can find a rabbi who will perform the wedding, I don't see why it couldn't be in a synagogue. But there is no requirement that a Jewish wedding be held in a synagogue, as a matter of fact, they usually aren't. Jewish weddings can be performed anywhere. Traditionally, they were performed outdoors, in the town square or other public place. Today they are performed in hotel ballrooms, in restaurants, in parks, in reception halls, in private homes, and other places. What makes a wedding Jewish is not its location, but that the officiant is a rabbi or cantor, that the Jewish blessings are said, the Jewish rituals are observed, and that other religions' traditions are not included. Please be aware that many rabbis will not perform intermarriages, and those who will usually have different criteria under which they will agree to perform the wedding. A Jewish wedding involves blessings and readings that celebrate the creation of a new Jewish family and establishment of a new Jewish home. I suggest before you and your girlfriend decide to get married, you have some serious conversations about the place Judaism and your religious tradition, if any, will have in your lives together. Your wedding should be a reflection of who you both are as individuals and who you intend to be as a married couple. I hope this helps.