How to join values from a R dataset to a shapefile of polygons I've created in QGIS?

This question would be easier to answer, and you would probably get quicker and more targeted answers if you include example data with the question.

This is a task you can accomplish readily in either R or QGIS. QGIS is more user friendly for making maps and one-off tasks, it is menu-driven and you see the results of your manipulations as you go. R is known for having a tough learning curve, but is more suited to automating tasks so you can do them repeatedly, and is more transparent, reproduce-able, and easier to audit or check for errors.

In either case, you need a key column in your two data sources that you can use to join by.

Assuming your count data is in a table that includes a column called "Area" and one with counts called "Turtles", and your polygons are a shapefile called "Zones_Example.shp" that includes a column called "Zone".
In R you can use the sf package to read in the shapefiles and the table, then use the dplyr/tidyverse package to left_join the count table to the shapefile.

library(tidyverse)
library(sf)

### Creating example tabular count data
countdata = tibble(Area = factor(1:4),
                   Counts = c(30, 4, 15, 20)
                   )
countdata
# # A tibble: 4 x 2
#    Area  Counts
#    <fct>  <dbl>
# 1     1     30
# 2     2      4
# 3     3     15
# 4     4     20


###  Reading in shapefile with zones
Zones = st_read("Zones_Example.shp")

Zones
# Simple feature collection with 4 features and 2 fields
# geometry type:  POLYGON
# dimension:      XY
# bbox:           xmin: 1086463 ymin: 439041.6 xmax: 1159460 ymax: 494408.7
# epsg (SRID):    NA
# proj4string:    +proj=lcc +lat_1=48.7333333333333 +lat_2=47.5 +lat_0=47 +lon_0=-120.833333333333 +x_0=500000.0001015999 +y_0=0 +datum=NAD83 +units=us-ft +no_defs
#   id Zone geometry
# 1  1    1 POLYGON ((1087565 491745.9,...
# 2  2    2 POLYGON ((1117774 494408.7,...
# 3  3    4 POLYGON ((1118233 458966.4,...
# 4  4    3 POLYGON ((1127506 462639.2,...

###   This pastes the columns from the "countdata" table into the "Zones" shapefile table 
###    If your key columns have the same name, the `by= ...` arguement isn't strictly necessary  
Zones = left_join(Zones, countdata, by = c("Zone" = "Area"))

### Notice new column "Counts"
Zones
# Simple feature collection with 4 features and 3 fields
# geometry type:  POLYGON
# dimension:      XY
# bbox:           xmin: 1086463 ymin: 439041.6 xmax: 1159460 ymax: 494408.7
# epsg (SRID):    NA
# proj4string:    +proj=lcc +lat_1=48.7333333333333 +lat_2=47.5 +lat_0=47 +lon_0=-120.833333333333 +x_0=500000.0001015999 +y_0=0 +datum=NAD83 +units=us-ft +no_defs
#   id Zone Counts                       geometry
# 1  1    1     30 POLYGON ((1087565 491745.9,...
# 2  2    2      4 POLYGON ((1117774 494408.7,...
# 3  3    4     20 POLYGON ((1118233 458966.4,...
# 4  4    3     15 POLYGON ((1127506 462639.2,...

st_write(Zones, "Zones_Example.shp", delete_dsn = TRUE)  # delete_dsn = TRUE overwrites the file

One of the keys to making this work is to be sure the values and data type you are joining on match.

Displaying this in QGIS, you style by the new "Counts" column in the Zones_Example layer: styling toolbar >> Select "Graduated" from styling dropdown >> select your "Counts" attribute from value dropdown >> Choose a color ramp and click the "Classify" button near the bottom of the styling panel

You can do the same thing entirely within QGIS using the MMQGIS plugin. This plugin has an option to "Attributes join from csv files" under the combine menu. You choose your csv file and select the columns from each table to match and set output.

You can load polygons as a simple feature (sf) object, then join your table of counts and plot and shade using the turtle counts.

Reproducible example below.

library(ggplot2)
library(sf)

# Basic shapes (two squares) and an associated turtle table.
shapes <- data.frame(polyid = 1:2, 
                     wkt = c("POLYGON((0 0, 0 1, 1 1, 1 0, 0 0))", 
                       "POLYGON((2 0, 2 1, 3 1, 3 0, 2 0))"))
polys <- st_as_sf(shapes, wkt = 'wkt', crs = st_crs(4326)) # make sf

turtles <- data.frame(location = c(1, 2), 
                      turtles = c(20, 5))

# Join the turtle data to the polygons
shapes <- merge(polys, turtles, by.x = 'polyid', by.y = 'location')

# Plot using the 'turtles' column to shade
m <- ggplot(shapes) 
m + geom_sf(aes(fill = turtles))