I would say: "The strategy impacts the service quality."
According to prescriptive grammar (which as we know always lags a good few decades or so behind actual usage), this has been an incorrect way to use the verb "impact," which (according to the rules) is a physical experience - a meteor can impact the surface of the moon, but a rude word cannot impact the way that you feel. However, just as with many other words in English, "impact" has been accepted to the realm of metaphorical expression: it is pretty easy to see the relationship between the effect that a meteor has when it hits an orbiting body, and the effect that cruelty has on our emotional state. In either case the affected object is/feels acutely changed by being on the receiving end of the action. If this is the idea you wish to communicate, then I think "impact" really is the best choice of word here. However, while "impact" as a noun is frequently followed by a preposition - "there was a heavy impact on the roof," or "we heard a loud impact over our heads," - when used as a verb, impact does not require a prepositional phrase; it requires a direct object. Therefore your sentence structure would be (noun phrase/subject) "The strategy" (verb phrase-transitive) "impacts" (noun phrase/direct object) "the service quality."
I realize that in the sentence given by the OP, "impact" is used as a non-count noun (which I am not sure about as general practice, but it seems like it could be argued that this is valid usage). However, in looking at the sentence structure that would result from this, we get (noun phrase/subject) "The strategy" (verb phrase-transitive(noun phrase/direct object) "generates impact" (prepositional phrase(object of preposition) "over the service quality." This implies that the impact generated by the strategy is superior to/more than that of the service quality, which I do not think is the intention. It seems that the change of state of the service quality in response to the strategy is what is meant to be described.
As to the question specifically of "over" versus "on," the above ambiguity arises mostly from the choice of "over" as the preposition. Changing it to "on" removes the ambiguity of whether the service quality is being affected or having an independent effect. However, either prepositional phrase gives the impression of a surface effect to the service quality, which is at odds with the core meaning of "impact."