Are there any examples of constructed languages that have verbs with different forms depending on the object of the verb?
I am not familiar with any “well-known” conlangs, but a quick search reveals that Klingon verbs inflect for both subject and object. This is a phenomenon called polypersonal agreement and it is very common in natural languages, and so it is only expected that it would also crop up in several conlangs. Among them, for example also my own, for which I just yesterday put up a puzzle that involves decoding the verbal system here.
I cannot currently think of any examples of a language which inflects for only the object, but it’s a rather low-hanging fruit and I cannot imagine that it has never been done before.
CALS lists 9 conlangs where the verb only agrees with P in transitives, 20 where it can agree with either A or P, and a further 127 where the verb agrees with both A and P, though of the ones listed, the only really well-known one is Klingon, which Adarian also mentioned. Including Mark Rosenfelder's langs (which aren't in CALS) in the at least relatively well-known bucket, Wede:i and Old Skourene both inflect the verb for both A and P to different extents (though note that Old Skourene works on an ergative basis).
Of the languages that mark P but not A, most of them are listed as having ergative agreement and so presumably agree with S as well, however two of them, 'Yemels and Snahhian are listed as accusative, and so presumably has the verb agree only with P.
Interestingly enough, despite conlangers often liking to solve problems by throwing affixes at them, a complete lack of personal agreement is actually overrepresented in CALS, when compared to WALS.
Әřant has a change that, while it may not be what you're asking for, does fit the question in that depending on the animacy valency of the object, the verb conjugates differently:
men xshanni - - tassu han-ne!
it drop.2S.IMPER - - thou.2S INTERJ-hey.EMPH
san xshannos - - tassu han-ne!
her drop.2S.INDIC - - thou.2S INTERJ-hey.EMPH
The person has a higher animacy valency, so requires a (slightly) more polite idiom.
I believe your question conflates two related, but crucially distinct concepts, patients (a thematic relation and semantic concept) and objects (a verbal argument and syntactic concept).
By conventional syntactic definition, if only one argument is being agreed with, that argument is the subject. In that sense, it's by definition impossible for a verb to agree only with an object.
However, the patient of a verb does not have to be its object! In ergative languages, it's generally acceptable to analyze that the syntactic subject of a transitive verb is its patient.
In summary: it makes no sense to say that a verb agrees only with its object, because it would causes that object, by definition, to be the subject, merely reversing the agency alignment of the verb (cf. English I like X vs. Spanish me gusta X).