No. It would take several years or longer before the garden soil would test organic.
Weed treaters are among the worst. They can remain in the soil for a long long time.
If you were to try and get your garden certified organic you would have to quit using the lawn chemicals for 3 years (as well as any other banned substances along with a soil building plan, a 4+ year crop rotation plan and many other requirements-see it all at) and than your garden would be eligible for organic certification.
But since you are a home gardener not going for organic certification organic can be what ever you want it to be. If you really want to be organic I suggest you read up on what organic gardening is really about and what it does for your soils, plants and local environment. all have a lot of information about what makes a garden or farm organic