I estimated a mixed model with the GENLINMIXED command in SPSS. In the model options I specified that I want to estimate simple contrasts for my interaction variable (category*group) by the variable "group". The output of the "simple contrasts" looks like this: 
I would like to get the same output in R where I estimate the model with glmer from the lme4 package. Does anyone know how to do that?
In general you would use the emmeans package for this. Here's an example using lm() and a built-in data set: it should work equally well for a glmer model, substituting ~group | category in the emmeans() statement and ref="0" (or something like that) in the contrast() statement.
library(emmeans)
warp.lm <- lm(breaks ~ wool*tension, data = warpbreaks)
warp.emm <- emmeans(warp.lm, ~ tension | wool)
cc <- contrast(warp.emm, "trt.vs.ctrl", ref = "L")
summary(cc, infer = TRUE) ## infer = TRUE to get CIs
You can choose the method of multiple comparisons correction using the adjust argument to summary(); I think you might use adjust = "tukey" to match your SPSS results, but I'm not sure.
wool = A:
contrast estimate SE df lower.CL upper.CL t.ratio p.value
M - L -20.556 5.16 48 -32.4 -8.74 -3.986 0.0005
H - L -20.000 5.16 48 -31.8 -8.19 -3.878 0.0006
wool = B:
contrast estimate SE df lower.CL upper.CL t.ratio p.value
M - L 0.556 5.16 48 -11.3 12.37 0.108 0.9863
H - L -9.444 5.16 48 -21.3 2.37 -1.831 0.1338
Confidence level used: 0.95
Conf-level adjustment: dunnettx method for 2 estimates
P value adjustment: dunnettx method for 2 tests