Let me first formalize the problem:
Cluster edition
Instance: A graph $G$ and an integer $k$.
Question: Can $G$ be transformed into a cluster graph by deleting at most $k$ edges?
It is not difficult to see that the problem is, already for general graphs, fixed-parameter tractable (FPT) parameterized by the number of edge deletions $k$. Being FPT means the problem is solvable in time $f(k) n^{O(1)}$, where $f$ is a computable function depending solely on $k$.
The observation is the following: cluster graphs are precisely the $P_3$-free graphs, i.e., the graphs that don't contain the path on 3 vertices as an induced subgraph. So, in order to produce a cluster from $G$, the $k$ edge deletions must "hit" each $P_3$. Clearly, there are two ways to destroy a $P_3$: either we remove edge $a$ or edge $b$. The algorithm is as follows: as long as there is a $P_3$, branch by choosing either one of its edges to delete. This will give you a straightforward $2^k n^{O(1)}$-time algorithm, that will be quite practical if $k$ is not too large.
Looking up the literature, the problem can be solved in $O^(1.77^k)$ time by a more clever yet simple branching algorithm (see <PERSON> et al. [1]). Here the $O^$ notation hides polynomially bounded factors.
It is interesting to ask whether you can do better for chordal graphs, or indeed $q$-trees (I'm using $q$ here to make a distinction between $k$). 603
The problem is solvable in polynomial-time for cographs [2], but NP-complete for chordal graphs [3]. I don't know if the complexity for $q$-trees is known, and I also didn't think hard about it but merely glanced the literature.
[1] <PERSON>, <PERSON>, <PERSON>, <PERSON>, and <PERSON>. "Graph-modeled data clustering: Fixed-parameter algorithms for clique generation." In Italian Conference on Algorithms and Complexity, pp. 108-119. Springer Berlin Heidelberg, 2003.
[2] <PERSON>, <PERSON>, <PERSON>, and <PERSON>. "The cluster deletion problem for cographs." Discrete Mathematics 313, no. 23 (2013): 2763-2771.
[3] <PERSON>, <PERSON>, <PERSON>, and <PERSON>. "Complexity of the cluster deletion problem on subclasses of chordal graphs." Theoretical Computer Science 600 (2015): 59-69. 603