There are two layers underlying the issue: The first layer is, formally we can show every classical conservation law has a quantum counterpart through, say, canonical quantization(see Weinberg's QFT chap7). The second layer is, if we go beyond the formal level, it is really that we would want the conservation law to hold and strive to make it hold in the quantized case too, unless there are obstructions such as quantum anomalies.
On the surface of the issue, if we canonically quantize a classical Lagrangian, all conservation laws should be preserved, and this is fairly simple to see: the field operator equations have exactly the same forms of their classical versions, hence if any classical quantity has a vanishing 4-divergence, so will its quantized version.
However, reading between OP's lines, I suspect OP is having the misconception that a QFT can be defined by "a Lagrangian + a quantization recipe(canonical, path integral etc.)" This is not entirely true, a regularization and renormalization scheme must also be specified to make the QFT unambiguous. During this procedure, many symmetries are broken in the intermediate stages, and it is not a priori clear that the symmetries can all be restored after the procedure is finished.
The common practice is, through carefully controlling the regularize-renormalize procedure, we define a QFT preserving as many symmetries and conservations as possible. It is not always possible to preserve all the classical symmetries at hands, in such cases we will have the quantum anomalies.
Hence my earlier comment under your question:
My impression is that they will still be conserved separately in the quantized case, unless there are anomalies.