No, but you will develop new veins for the same reason you developed the veins that have been treated. The treated veins are gone for good but new reticular veins and associated telangiectasia (spider veins) will develop over time.
The time for new veins to develop depends mainly on factors such as family history, pregnancy and occupation. It is expected that a forty year old lady would have a “vein free” period of somewhere around 7 years after treatment before she feels the need for a bit of a “tidy up”.
Younger people can expect around 5 years (might be worse if have several pregnancies over this period) and older patients up to 10 years between treatments.