There are two ways, I believe:

1. Show that doing a nodal slide across the central fiber is same as performing a zero-area Lagrangian surgery of the fiber with a disc given by parallel a circle transporting to the singular fiber.
2. I would guess that this is what Vianna had in mind. Vianna shows that performing three rational blowdowns (loosely, consider rational blowdown to be a nodal trade, but this actually changes the manifold since you're doing it at a non-Delzant corner) $\mathbb P (a^2,b^2,c^2) $ is a standard $\mathbb P ^2$ when $(a,b,c)$ is a Markov triple (Corr. 2.5 in the paper). Let $T(a,b,c)$ be the central torus fiber in the orbifold $\mathbb P (a^2,b^2,c^2)$, thus it is monotone. Now, by abuse of notation, call the torus obtained in $\mathbb P^2$, after performing the three rational blowdowns, $T(a,b,c)$, which is the Vianna tori corresponding to the Markov triple $(a,b,c)$. Now you check that performing a rational blowdown preserves monotonicity to finish off.

There are two ways:

1. Show that doing a nodal slide across the central fiber is same as performing a zero-area Lagrangian surgery of the fiber with a disc given by parallel a circle transporting to the singular fiber.
2. I would guess that this is what Vianna had in mind. Vianna shows that performing three rational blowdowns (loosely, consider rational blowdown to be a nodal trade, but this actually changes the manifold since you're doing it at a non-Delzant corner) $\mathbb P (a^2,b^2,c^2) $ is a standard $\mathbb P ^2$ when $(a,b,c)$ is a Markov triple (Corr. 2.5 in the paper). Let $T(a,b,c)$ be the central torus fiber in the orbifold $\mathbb P (a^2,b^2,c^2)$, thus it is monotone. Now, by abuse of notation, call the torus obtained in $\mathbb P^2$, after performing the three rational blowdowns, $T(a,b,c)$, which is the Vianna tori corresponding to the Markov triple $(a,b,c)$. Now you check that performing a rational blowdown preserves monotonicity to finish off.