No, mescaline only has the one, single carbon-nitrogen bond. Mescaline cations form through protonation at nitrogen. Methylation leads to different molecules - N-methylmescaline, trichocerine and the N-methyltrichocerine quaternary cation. Both N-methylmescaline and trichocerine may (also) be protonated - reversibly - to form cations, but only the N,N,N-trimethylmescalinium [N-methyltrichocerinium, N-methyltrichocerine quaternary cation] counts as a quaternary ammonium compound.

• mescaline is a primary amine
• N-methylmescaline is a secondary amine
• trichocerine is a tertiary amine

Only these compounds in the series have the spare lone pair on the nitrogen atom for reversible protonation to occur. Removal of a methyl group from a quaternary ammonium cation requires a more vigorous chemical process which needn't be discussed here.

[URL unfurl="true"]https://en.wikipedia.org/wiki/Reinecke's_salt[/URL]

This can be used to precipitate alkaloids as a convenient analytical reagent. Perhaps the quaternary ammonium compounds are selected by being far less soluble.

Hm, "water soluble base" is a bit vague, on reading the paper we see:

"Weak" does rather imply that these weren't the quaternary ammonium compounds, which form strong hydroxide bases due to the lack of a labile proton.

[As you may also be aware, Desmodium triflorum, as per the paper which you attached, has since been renamed Grona triflora]

[URL unfurl="true"]https://en.wikipedia.org/wiki/Grona_triflora[/URL]