No, there is no such direct link between the budget deficit and the trade deficit. The following graph shows exports, imports, and trade deficits (exports minus imports) since 1930:
The actual numbers and sources are at http://home.att.net/~rdavis2/tradeall.html. As can be seen, the trade deficit has increased fairly steadily since 1991, even during the budget surpluses from 1998 to 2001. The only relationship that I recall hearing about is that the budget deficit provides additional government debt in which foreign countries can invest the excess dollars that they receive via the trade deficit. In fact the trade deficit may help keep interest rates low by creating more dollars competing for that government debt. Of course, there is no free lunch. That "benefit" comes at a cost.