No, an element of [itex]\mathbb{Q}[/itex] is defined as a fraction [itex]\frac{m}{n}[/itex], where n is nonzero. So you can't take 0 is the denominator, by definition.

Whether we're discussing the Integers, the Rationals, the Reals, or Complex Numbers (all with the usual arithmetic operations):

It's always true that 0 times any element is 0, 0 being the identity element for the addition operation. Because of this, there is no multiplicative inverse for 0, and thus division (the operation that is the inverse of multiplication) by 0 is undefined. Therefore, when discussing whether division is closed, we exclude the case of division by zero.