Determine the length of the largest contiguous segment
Code Review

PEP 8

PEP 8 recommends snake_case for variable names, so N, Q, A, L, R, and D all violate the guidelines.

A space should be used around operators: index - 1 and R - 1

Unused variables

N and testcase are never used, so should be replaced with the throwaway _ variable.

Variable naming

N, Q, A, L, R, and D are all "almost meaningless".  Yes, they are directly from the problem description, but you could do better naming them array_size, num_queries, array, left, right and difference.

Is count_length a count of lengths?  Doesn't seem to be; it is more like a max_length.

Is count_temp a count of temporaries?  Or is it more like a segment_length?

_, num_queries = [int(x) for x in input().split()]

array = list(map(int, input().split()))

for _ in range(num_queries):
    left, right, difference = [int(x) for x in input().split()]

    if left == right:
        print("1")
        continue

    max_length = 0
    segment_length = 0

    for index in range(left, right):
        if array[index] == array[index - 1] + difference:
            segment_length += 1
            if index == right - 1:
                segment_length += 1
                max_length = max(max_length, segment_length)

        else:
            segment_length += 1 # to include the first element of AP
            max_length = max(max_length, segment_length)
            segment_length = 0

    print(max_length)


Optimization

Without changing the Algorithm???

Ok, you're looking for just key-hole level optimizations.  Let's see what we can do.

    for index in range(left, right):
        if array[index] == array[index - 1] + difference:
            ...
        else:
            ...


The above is inefficient code.  Indexing into a list takes time, and you are doing it twice.  Moreover, subtraction by 1 takes time, and it was to get the value which was looked up on the previous iteration!

Compare the above code with:

    prev = array[left - 1]
    for curr in array[left:right]:
        if curr == prev + difference:
            ...
        else:
            ...
        prev = curr


The indexing has been replaced with iteration over an array slice, and the current value from one iteration is carried forward to as the previous value in the next iteration.  That should be a win, performance wise.

    segment_length = 0

    for ...:
        if ...:
            ...
            if ...:
                segment_length += 1
                max_length = max(max_length, segment_length)

        else:
            segment_length += 1 # to include the first element of AP
            max_length = ...
            segment_length = 0


Why are you starting the count from 0?  That means you need this segment_length += 1 to account for that first element all the time.  If you initialized segment_length = 1, you can skip that adjustment statement every time you reach the end of an arithmetic progression, saving another wee bit of time:

    segment_length = 1

    for ...:
        if ...:
            ...
            if ...:
                max_length = max(max_length, segment_length)

        else:
            max_length = ...
            segment_length = 1


Finally, the code I hate the most:

    for index in range(left, right):
        if ...:
            ...
            if index == right - 1:
                ...

        else:
            ...
        ...


When does the for loop end?  After index loops the final time with index = right - 1, of course.  So the only time this if condition will be true is during the last iteration of the loop.  During all of the remaining loop iterations, you are wasting time, subtraction 1 from right and comparing the result to index, when it can't possibly ever be true.  Simply move the code to follow the loop.

_, num_queries = [int(x) for x in input().split()]

array = list(map(int, input().split()))

for _ in range(num_queries):
    left, right, difference = [int(x) for x in input().split()]

    if left == right:
        print("1")
        continue

    max_length = 0
    segment_length = 1

    prev = array[left - 1]
    for curr in array[left:right]:
        if curr == prev + difference:
            segment_length += 1

        else:
            max_length = max(max_length, segment_length)
            segment_length = 1

        prev = curr

    max_length = max(max_length, segment_length)

    print(max_length)


With changing the Algorithm

You explicitly requested optimization without changing the algorithm, so I'll leave you to discover where there is an algorithmic improvement to be made.