It is best to think about this problem discretely. The clock is divided into 60 positions where the two hands could possibly coincide. Lets call the start time the 0th second, where all the hands are coinciding.
Unfortunately there are two different types of clock. In one type, the minute hand moves smoothly with the second hand. Lets consider this case first.
After 60 seconds the minute hand has moved to position 1. This means that after 60 seconds there has still only been 1 counted coincidence (the starting position). After 120 seconds there has been a second counted coincidence, and the minute hand has moved to position 2. This pattern continues for 3540 seconds without deviation and we have 59 counted coincidences, with the minute hand on the 59th position and the second hand on the 0th position. We note that in 60 more seconds we will have the starting condition again; so after a second iteration of the above, we will have 118 counted coincidences, with the minutes hand on the 59th position and the second hand at the 0th position. After 60 more seconds, we have our final coincidence at exactly 4pm, and the final count is 119.
However, there is a second type of clock where the minute hand stays perfectly still until the second hand strikes the 0th position, after which point, the minute hand jumps to the next position. On this type of clock, the two hands coincide at the 0th second, then the minute hand jumps to position 1, and the two hands coincide again at the 1st second. We then follow the logic of the previous case, and we find that each iteration offers exactly one more coincidence. Since we have two iterations to consider, we have a final count of 121. (And at 4pm and 1 second we would have a count of 122, but this is outside of the range of counts). If your book offers 121 as an answer, it is probably thinking about this style of clock.
PS. It also depends on what is meant by 'between': usually this either includes both endpoints or it doesnt. If the coincidences at exactly 2pm and 4pm are disallowed, then either 117 or 119 are acceptable, depending on the type of clock.