What is quasi-aromaticity?
In general, we can describe a quasi aromatic compound as a compound, which is ionic in nature with a counter ion, and the $\pi$ electrons in such compounds follow Huckel's rule ($4n+2$).

In other words, quasi aromatic compounds are those in which the charges present on the molecule are a part of aromaticity of the compound. A few examples of such compounds are depicted in the diagram:



However, it has a deeper and broader meaning. For a more in-depth explanation, please read the given references.



References:


T. M. Krygowski, B. Bankiewicz, Z. Czarnocki, M. Palusiak, “Quasi-aromaticity—what does it mean?” Tetrahedron 2015, 71(30), 4895–4908 (https://doi.org/10.1016/j.tet.2015.05.074).
E. Kleinpeter, A. Koch, “Characterization and quantification of quasi-aromaticity by spatial magnetic properties (TSNMRS),” Tetrahedron 2015, 71(33), 5275–5284 (https://doi.org/10.1016/j.tet.2015.06.019).