**TL;DR**: *Tree-level* flavour changing neutral currents are not allowed in the Standard Model. There could be, if you invent another particle or two (and indeed there are New Physics proposals about this). <br> But the Standard model *does* allow for flavour changing neutral currents with higher order diagrams, like *box* or *penguin* diagrams.

**More in depth**

A "flavour changing current" is a process featuring the violation of quark flavour conservation. When mediated by a *neutral* particle such as the $Z^0$ boson (or a photon), they do not entail a net charge exchange and are hence called "flavour changing *neutral* currents".

> if a top quark emits a Z boson or a Higgs boson

At tree level the $Z^0$ boson *cannot* allow flavour mixing.

This is because the $Z^0$ boson has no charge and can hence only couple quarks of the same charge: so both $i$ and $j$ have to be $\in (u, c, t)$ or $\in (d, s, b)^\dagger$. You can show, then, the matrix element for the tree level interaction goes as :
$$ \mathcal{M} \quad \propto \quad g_W \cdot(u, c, t)  \left (\begin{array}{c} \bar u \\ \bar c \\ \bar t  \end{array} \right ) \quad \propto \quad u\bar u + c \bar c + d \bar d,$$
i.e. no cross-generational coupling and hence no flavour mixing.

If the process were mediated by a $W^{\pm}$ boson, on the other hand, you'd have sometime like:
$$ \mathcal{M} \quad \propto \quad \bar u\bar V_{ud} d + \bar u V_{us} s + ...,$$
i.e. cross-generational coupling and hence flavour mixing. <br> $V$ is the [CKM][1] matrix.

Hence, the lowest order diagrams for Flavour Changing Neutral Currents in the Standard Model are *box* or *penguin* diagrams:
[![enter image description here][2]][2]

The *penguin diagram* gets his name because:

[![enter image description here][3]][3]

--------------

$\dagger$: There are some subtleties about the mass eigenstates $(d,s,b)$ not being the same eigenstates of the weak force $(d',s',b')$. The two bases are related by the CKM matrix. I ignore all of this, just to provide a qualitative picture.


  [1]: https://en.wikipedia.org/wiki/Cabibbo%E2%80%93Kobayashi%E2%80%93Maskawa_matrix#CKM_matrix
  [2]: https://i.sstatic.net/d8xCs.png
  [3]: https://i.sstatic.net/PfXw2.png