Should the distribution be normal?
Yes, for the t-statistic to have a t-distribution, the data should be normal; in practice, it won't ever be normal - a price is a strictly non-negative quantity for starters, but generally people hope that approximate normality is sufficient for the purpose of obtaining p-values that are close enough.
If so, can I just construct a Q-Q plot to test for this?
That's a good diagnostic check on the reasonableness of the assumption, but it won't tell you that your data is normal, only that it's not so severely non-normal as to cause undue concern.
Distributionally, your main worry here will tend to be heavy skewness, so you would worry most about seeing a strong curve in a Q-Q plot. Practically, prices tend to be somewhat right-skew, but in this case probably not heavily so. (The t-distribution is also affected by symmetric heavy or light tails but the main effect is to push the significance level up or down a little bit; in general people feel they can tolerate a moderate effect on that).
There are further assumptions - apart from a possible shift in the mean, the distributions the observations are drawn from are the same (basically, with the assumption of normality, this boils down to equality of variance), independence (which the 'random sample' is often considered to take care of, but in some situations won't give it to you; even though it's probably not quite true, it's probably good enough here, but see the comments below that suggest some possible sources of dependence even so; the question is whether these sorts of things are enough to cause a problem).