No. In BQE CORE, you can import budgets into forecasting, but not the other way around. In the Allocation & Forecasting screen, you can use More > Copy from Budget to bring budgeted time and expenses into your forecasting allocations. However, CORE does not support importing forecasting data into budgets.
Budgets can only be created or imported from budget templates or existing projects (via the Projects > Budget tab). Check CORE Help Center for details.