The comments provided by the commenter both solved my problem but with slightly different results. For the benefit of other users these differences are illustrated below.
Solution 1:
Here the markers appear in the legend twice for each series.
Code:
import seaborn as sns
fmri = sns.load_dataset("fmri")
ax=sns.lineplot(data=fmri, x="timepoint", y="signal", hue="event", marker="o")
for h in ax.legend_.legendHandles:
h.set_marker('o')
Output:

Solution 2a:
Here the markers appear in the legend once for each series. The line style is different for each series.
Code:
import seaborn as sns
fmri = sns.load_dataset("fmri")
sns.lineplot(data=fmri, x="timepoint", y="signal", hue="event", style="event", markers=["o"]*2)
Output:

Solution 2b:
Very much like solution 2a, but in order to have all lines in "solid" style the following option is added:
dashes=[""]*2
Code:
import seaborn as sns
fmri = sns.load_dataset("fmri")
sns.lineplot(data=fmri, x="timepoint", y="signal", hue="event", style="event", markers=["o"]*2, dashes=[""]*2)
Output:
