I believe that indeed equilibrium/nonequilibrium here refers to thermal equilibrium/nonequilibrium. Informally, a system is in equilibrium if you leave it be and wait long enough, so that all currents (of energy, particles, etc) flowing through the system quiet down. Take a glass of water and wait, and it equilibrates with its surrounding. However, if you put the water in a pot and heat from a stove below while the top is being cooled by air, then even if the water reaches a steady state where you cannot see any macroscopic motion, energy is flowing from the bottom to the top, and this steady state is a nonequilibrium one.
Turbulent states occur when you keep pumping energy in and it is continually being dissipated, so energy flows through the system (for example, you stir the fluid on a macroscpic scale, and energy "cascades" down to smaller scale eddies until at very small eddies it is being dissipated).
More precisely, the common definition of an equilibrium system is one that is statistically time-reversal symmetric: you cannot tell by looking at a video of the dynamics whether the video is going forward or backward. If you model your system as a Markov process (e.g., by using a Langevin equation), then equilibrium means that the process is "reversible", or satisfies the detailed balance condition. Turbulent fluids do not satisfy these conditions precisely because there is an energy flow.