How to decide if two conjunctive queries cannot have any result in common
Your example queries are not conjunctive queries in the sense I came to know them. On Wikipedia they also are defined differently. What you maybe are looking for is the boolean query evaluation problem. $Q_1(D) \cap Q_2(D)$ can be expressed as $(Q_1 \wedge Q_2)(D)$, thus deciding whether $Q_1(D) \cap Q_2(D)$ is empty is the same as deciding whether $(Q_1 \wedge Q_2)(D)$ has any solution at all.
For conjunctive queries (as defined by Wikipedia) there are algorithms, faster than the brute force method, for this problem, if the query has a certain form, for example acyclic conjunctive queries.