A Dyck path is non-decreasing if the y-coordinates of its valleys, which are local minima along the path, form a non-decreasing sequence. To count the number of subpaths (or subwords) within non-decreasing Dyck paths, we use generating functions and recursive relations. These techniques enable us to count subpaths of various lengths, such as two, three, four, and five, while considering both the length and the number of subwords. Using constructive recursive proofs, we establish a combinatorial interpretation of the counting process, expressing the results as linear combinations of Fibonacci and Lucas numbers. In several cases, we provide new interpretations of sequences from the On-Line Encyclopedia of Integer Sequences (OEIS).