No

If Q is at the first place then you have 7 choices for the second letter then 6 choices for the third one and finally 5 choices for the fourth one

That makes 7x6x5 = 210

But if Q is at the second place you have also 210 possibilities. The same if Q is at the third place and the same for the fourth place.

Therefore there are 210 x 4 = 840 words

Another way is to calculate the number of group of 4 letters containing Q

There are groups of 4 letters with Q (the same number as the number of groups of 3 letters among 7 letters)

For each group of 4 letters you can build 4! = 24 words

The total number of words is 35 x 24 = 840