According to my Cotton & Wilkinson, the chemistry of the vanadates is very complicated. It says that vanadium(V) oxide, V2O5, dissolves in sodium hydroxide to give colorless solutions which consist mainly of the VO43�?/SUP> ion at pH 13 and above. As the pH is lowered, the solution contains various species which change as the pH is lowered. At about pH 3 and lower, a major species is the VO2+ ion (still vanadium(V)).
When hydrogen peroxide is added to acidic (I wonder if this should be basic, see below*) vanadium(V) solutions, a red color is formed which results from peroxo complexes in which oxygen atoms in VO43�?/SUP> are replaced by one or more O22�?/SUP> groups. Several of these peroxovanadates have been isolated, including a complex of the V(O2)3�?/SUP> ion. The monoperoxovanadium(V) species, according to this, would have the formula V(O2)O33�?/SUP> (replacing one oxide ion in VO43�?/SUP> with the peroxide ion O22�?/SUP>), but it was not mentioned in Cotton & Wilkinson. Nor was the ion you mentioned, V(O2)O+.
*This might be an error in Cotton & Wilkinson, unlikely as that is. But if we go with the assumption that hydrogen peroxide results in replacing oxide ions one at a time with peroxide ions to the vanadium(V) species that is predominant in strongly basic and strongly acidic solution, then we have:
Basic Solution
VO43�?/SUP> (aq) + H2O2 (aq) ––�?gt; V(O2)O33�?/SUP> (aq) + H2O (l)
Acidic Solution
VO2+ (aq) + H2O2 (aq) ––�?gt; V(O2)O+ (aq) + H2O (l)