No, the biblical account of the Exodus does not place the Israelites' journey through the Syrian Desert. The Bible describes them crossing the Red Sea (or Sea of Reeds), wandering in the Sinai Peninsula, and eventually entering the Promised Land (Canaan, which is roughly modern-day Israel and Palestine).
The Syrian Desert is located further north and east, encompassing parts of Syria, Jordan, Iraq, and Saudi Arabia. While the Israelites may have been geographically close to the edge of the Syrian Desert at certain points, the narrative clearly places their wanderings in the Sinai.