My intuition is that, since the concentration of sodium within a cell is higher than the extracellular concentration, when conductance increases, this corresponds to the channel being open and means sodium will enter the cell. Is this correct? And what impact will this have on the membrane potential?
No, not quite. You are on the right lines, but incorrect in thinking that intracellular sodium concentration is higher.
At the resting phase of a neuron, there is less sodium inside the cell and more sodium outside the cell. An ATPase sodium potassium pump is constantly pumping 3 sodium ions outside the cell for every 2 potassium cells that enter the cell.
For voltage gated sodium channels to open, there must be a significant change in voltage, which can be triggered by the binding of neurotransmitter to receptors for example. This triggers the sodium channels open and sodium will flood into the cell due to a high concentration gradient (because there are more sodium outside than inside) but also due to an electrical gradient (the neuron is more negative on the inside; positive sodium ions will be attracted to the inside).
Because sodium is flooding into the cell, this will cause the membrane potential to become more positive (depolarised) and will eventually reach an action potential (30mV). After an action potential, the channels will become de-activated and shut. After a period of hyperpolarisation, the neuron will return to a resting state.