Using "connotation of" in an active manner?

Connotes can be used as a verb; that is, one could say "The object connotes magic" or perhaps "The object connotes being magical". (The constructions "object connotes" and "object has ... connotation" both are slightly metaphorical uses of connote, as connote ordinarily is a term applied to words, phrases, statements, rather than to things. As suggested in a previous answer, some may regard "The object connotes magic" as nonsense rather than as a meaningful sentence.)

Words you could substitute in place of connotes to obtain a less-problematic sentence include expresses, implies, indicates, suggests, signifies, symbolizes, hints at.

You wrote "... I want to say an object that suggests magic ..." So why don't you just say "This object suggests magic"?

You can simply write: The object has magical connotations.

The British Museum uses connotations in exactly that manner here:

The British Museum has six objects with magical connotations associated with the name of John Dee (1527-1608/9).

A connotation is "an additional idea or emotion that a word suggests to you, that is not part of its usual meaning."

What kinds of additional ideas or emotions does your object suggest? Magical ones. Yes, magical modifies connotations, but the magical connotations are those suggested by the object.