Can the first quartile equal the third quartile?

yes, if all the data is the same number; when the range is zero.

* * * * *

That is not true. You need 25% of the values to be small, then 50% identical values, followed by 25% large values. Then the lower (first) quartile will be the same as the upper (third) quartile. The inter-quartile range (IQR) will be zero but the overall range can be as large as you like.