Intention is a Nominalization derived normally from the verb intend plus the nominalizing suffix -tion. As with many nominalizations, it forms a periphrastic idiom for intend: have D intention where D is some Determiner, like an, any, no, some, the, etc.
Also as with many nominalizations from predicates, constructions used with the predicate can be used with periphrastic constructions made from the derived noun, like Equi Infinitive Complements, for mental predicates.
- He doesn't intend to kill his gecko. ~ He has no intention to kill his gecko.
- He is not inclined to mow his lawn. ~ He has no inclination to mow his lawn.
In addition, the noun intention can take a gerund complement describing the content of what the subject intends. Naturally, a preposition is required to link the noun with the gerund, and since the gerund describes the content of the mental state, it uses the common possessive of.
- He has no intention of killing his gecko.
This is a fact about intention, however; other nouns (derived or not) may or may not allow this construction (in fact, they likely won't, because this kind of individual constraint helps distinguish predicates). Thus, the following are ungrammatical:
- *He has no inclination of killing his gecko.
- *He has no desire of killing his gecko.
- *He has no plan of killing his gecko.