Legal Question in Wills and Trusts in Virginia

The answer is, no, unless what's called a testamentary trust is created in the will or already exists at the time of the testator's death(and there is a need for whatever reason to name a trustee).

The executor probates the will and administers the estate of the deceased testator while the trustee administers the referenced trust in behalf of the named beneficiaries. The executor's job is likely to end in no more than a couple of years whereas that of the trustee could go on for years (until the trust is terminated).