Is there such a usage with the word "wonder"?
- It wonders me whether you've been to Paris.
I know of "bothers me" and "amuses me".
Yes, it is possible and once valid, but this usage has long gone out of use. If you dig into Middle English and Early Modern English texts you might find many examples of similar constructions. It is considered so obsolete that you probably won't even find this usage in major dictionaries.
The OED has several relevant entries.
To affect or strike with wonder; to cause to marvel, amaze, astound.
a1561 But howe they durst presume, it wonders me therfore.
a1627 Of all passages in the story of Job, that one thing wonders me.
1627 It wonders me to hear the desperate inference.
1788 She alarms me sometimes for herself, at other times she has a sedateness that wonders me still more.
It would be a very unusual construction. As a native English speaker, it sounds highly unusual.
Such sentences are written in the passive voice. "It bothers me" means the thing (it) is doing the action of bothering. For some verbs, such as "bothers" and "amuses," we find it grammatically valid to say they do these actions. It is considered reasonable for a thing to do an action like bothering.
There is no clear rule in English as to which words can be used in this way. It is not clear what inanimate objects or events are permitted to "do" by this use of the passive voice. It also varies from language to language. In English we might say "I broke the glass," but in Spanish one would say something which translated more literally to "The glass broke itself." In Spanish, the verb "to break" may be used in this way, but in English it would be awkward.
"Wonder," however, is not one of those words that we permit to be used in this way. A native English speaker would not normally phrase it that way.
However, I will note that in most languages, all things are possible. Someone might use this construction in a poetic way to describe something that truly instills a substantial sense of wonder. A poet who lives by the sea might use the phrase "The sunset over the ocean wonders me." They would be more likely to use a different phrasing, "The sunset over the ocean fills me with wonder," but they may wish to convey something more profound. One of the skills of a poet is to be able to use phrases which might be grammatically wrong but use them in a way which still causes the right feeling in those who listen to the words.